Common definitions and formulas for [linear algebra]

matrices and vectors

Linear operation and Transpose

Matrix

Elementary transformations of matrices:

Ladder-shaped matrix

is defined as:

1. If there are 0 rows, it will be below

2. The first non-0-element column number of each non-0 row is strictly monotonically rising from top to bottom.

or the number of 0 on the left of each line rises strictly from top to bottom, until it is all 0.

Corner: The position of the first non-0 element in each non-0 row.

Simple ladder-shaped matrix:

3. The corner position element is all 1

4. The elements above the corner are all 0.

Each matrix can be transformed into a ladder-shaped matrix and a simple ladder-shaped matrix using elementary rows.

Matrix elimination method for linear systems

Re-solving the simple equation by using the same solution transform

Three kinds of same solution transformations

1. Swap the upper and lower positions of the two equations

2. Multiply a non-0-number C by one equation

3. Add multiples of one equation to another

The above reflected in the augmented matrix is three kinds of elementary row transformation.

Matrix elimination Element Method:

determinant Type

Calculation (0 descending order method)

Other properties of the determinant:

The law of Clyde

Matrix

Follow the law

1. Linear Properties

2. Operational and polynomial of n-order matrices

Elementary matrix and its role in multiplication

For the unit matrix, the matrix obtained by making an elementary transformation becomes the elementary matrix.

There are three types of elementary transformations:

The block rule of multiplication:

Two common cases:

Matrix equation and invertible matrix

Adjoint matrix

linear relationships and rank of vector groups

Linear meter out

Linear correlation

Maximal unrelated groups and rank

Linear Systems

The expression form of the equation Group:

Discrimination of the solution condition

Basic solution system and general solution

characteristic vectors and eigenvalues, similarity and diagonalization

Similar

two sub-type

[linear algebra] Common definitions and formulas